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Properties

Label 1-95-95.17-r1-0-0
Degree $1$
Conductor $95$
Sign $-0.982 - 0.187i$
Analytic cond. $10.2091$
Root an. cond. $10.2091$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.342 − 0.939i)2-s + (0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.173 − 0.984i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.984 − 0.173i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s − i·18-s + (−0.939 − 0.342i)21-s + (−0.984 + 0.173i)22-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.173 − 0.984i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.984 − 0.173i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s − i·18-s + (−0.939 − 0.342i)21-s + (−0.984 + 0.173i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.982 - 0.187i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ -0.982 - 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1767802170 - 1.868072433i\)
\(L(\frac12)\) \(\approx\) \(0.1767802170 - 1.868072433i\)
\(L(1)\) \(\approx\) \(0.9223565861 - 0.9931757242i\)
\(L(1)\) \(\approx\) \(0.9223565861 - 0.9931757242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.984 - 0.173i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.342 - 0.939i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.98135770518217531849514316454, −29.62812808914594419659100819449, −28.09104119273540869789990167435, −26.91429010043483726294250617971, −25.92172520323100338441854735468, −25.407476203211366458521523823429, −24.37425853069745485940614474263, −23.19992730323075993973255197939, −22.01392563004468109425444409266, −21.19720271091912799640121382338, −19.71348438815784601881966168497, −18.76154287650181773569324831111, −17.418043975058359146609800650165, −16.10412777332303770928350032251, −15.22478334540377254566419336291, −14.46433164496284062734975852764, −13.087480482507805945294002787446, −12.44098821233523486340195659076, −9.97829340533415089742588057409, −9.1514200243059098579517436602, −7.86017759675165875025532754001, −6.86568591240963037166991393461, −5.27218632101250518940891513829, −3.89119442824375811129816938206, −2.55597237984191897337141684415, 0.66769473517189660564445617831, 2.61136917444070395285106627891, 3.41231115360505481604069947558, 4.948119818026468540083078003138, 6.80971678393313520431257633691, 8.375271067209906133229225270198, 9.61021203171702628485831432351, 10.43667545784786827476793935019, 12.08531733577547896342186510867, 13.165931551949986314580798249970, 13.88806909531061409519570712478, 15.007569792172347794277779056984, 16.4244020102644219776892076580, 18.198270858395307907851027038190, 19.16426091763895587643733981010, 19.86815435623128000085061756300, 20.84607498271861998889276216975, 21.85294329714583465939167076019, 23.018794871997052769039860065100, 24.1190196008869760289567351059, 25.26897192213336807253471232055, 26.70738251597767160050846473916, 27.07796898200707536148837042582, 28.90450713008216150070872378823, 29.48330036745819583981651496752

Graph of the $Z$-function along the critical line